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0 and II g 11, > 0. Now, applying Lemma 3.1 with
we obtain
If(t)g(t) I ~ 1 I f(t) I" 1 I g(t) I" 11/ll,llg 11, """'P (11/11,)" + q (II g 11,)"
(2)
This gives that/gEL1, and by integrating, we find that
~
1
1
1!/ll,llgll, ~p+ -q- 1• Hence (3) Equality in (2) would occur if «.- {l, and, consequently, in (3) if «. .. fl holds a.e. In other words, if
II g 11: I flt) I• •II/II~ I g(t) 1"·1 Note. The inequality (3) is homogeneous, i.e., it holds for qf and bg with a, bE~ wbel;aever It is so;forJanCI g.
210 Lebesgue Measure tl1ld Integration
The Riesz-Holder inequality implies that if /ELP and gEL1 , (1/p)+(l/q)•l then /gE£1. It is not true in general that the product of two integrable functions is also an integrable function. In fact, we have the following.
Remark.' The condition that p and q are mutually conjugate exponents cannot be dropped in Theorem 3.2, see Problem 3. For the converse of Theorem 3.2, see Problem 14. 3.3 Theorem (Riesz-Millkowsld Jaequality). Let 1 ~p ~ oo. Then for every pair/, gELP, the following inequality holds: 11/+g 11, ~ 11/11,+
II g IIJ..
Proof. The case for p = 1 is straightforward. If p • oo, we note that
{ =>
I /I ~ 11/11... a.e. I g I ~ II g 11... a.e. II+ g I ~ 11/11... +II g 11 ... a.e,
and hence the result follows in this case also. Thus, we now assume that 1
0 such that ·
11/llcr :E;; Kll/11,.
c
L', and there exists a
V/EI!.
Proof. Clearly, the result holds good for each (0 < q < oo; when .
.
.!!.
p= oo. Thus, assume that 0 < q
..> 1 and choose p. such that(1/"A)+(1/p.)= I. Then
J: Ill•· J: Ill'· :E;;
.
(t
I
If ~-1
r/Au:~ t~ 1
- (f~!l~}"'
Thu8/E L' which proves that U then
c L'. Further, if we setK"'=b-a,
111 n, :E;; Kll/11,1
The Lebesgue L' Spaces 219 Remark. The result in Theorem 6.1 holds good for any measurable set E with m(E)
6.% Example. Take E=]1, co[ and define afunctionj: E-+ R by f(x)=x-1/t, Clearly
f
E LP(E) if p
> q while f
1 ~ q < co. ~ L'~(E).
6.3 Theore& LetO
n L",thenjELrforall
Proof. For each r, q < r
t, 0 < t
< p, we can :6n4 a
<
1, such-that
r=tq+(1-t)p.
Now jELP =>
n L" =>/E.LPandjEL"
I1\JJ(l-t) E
£1/(1-t)
atid
If \Ill E
£1ft.
Moreover, we nc:»te· that 1/t > 1 and the exponents 1ft and 1/(1- t) are congugate to each other. Therefore, by Riesz-HOlder inequality (cf. Theorem 3.2),'we have
I J\r= \J\Iil\J\(1-t).p E which verifies that
f
£1,
E Lr..
The following theorem will be of considerable value as we continue our investigation on the properties of Lebesgue integrable functions. 6.4 Theorem. Let f E L', 1 ~ p < co and E > 0. Then there exists: (a) A step junction +such that Ill-+ < E. fh)\t continuous function g such that Ill- g II, < E.
n,
6.5 Corollary. (a) The space C[a, b] of all continuous junctions -+ R is dense in L' for each p(1 ~ p < co) and (b) The family of all step functions defined on [a, b] is dense in L' for each p(l ~ p < co).
f: [a, b]
Remark. If we replace the interval [a, b] by an infinite interval or more generally by any measurable set E with m(E)= co, then a function/ E L' may not be approximated by a continuous function. Note, for instance, that a constant function on R is continuous but not integrable. As such, the space C(E) of continuous functions on Eneed not even be a subset of Ll(E) or I!(E). while for approximation of f by step functions we have Theorem 6.6.
220 Lebesgue Measure and Integration 6.6 Theorem. Let [EL' (R}, 1 ~ p < oo and exists a step function p on R such that Ill- pll, <
E
>
0. Then there
E.
6.7 Corollary. The set of all step functions on R is dense in L'(R) for eachp, 1 ~ p < oo.
6.8 Theorem. For 1 ~ p
<
oo, the space L' is separable.
Proof. Consider the collection !R. of all step functions having discontinuities only at rational points in [a, b] and assuming only rational values. Clearly !R. is a countable set. But each step function can be approximated by a step function in !R. with respect to the norm 11·11, and also the family of all step functions is dense in £' (cf. Theorem 6.4). Thus, the countable family !R. is dense in £'..Hence I! is a separable space.l Remark. The situation is different when p = oo. 6.9 Theorem.
LIXJ is not a separable space.
Proof. We note that
II X[8,c]- [X•,d] l tXJ = 1, =>
Stp.{X£•,cl) n Stp.(Xr..dl) = p, .
d c ::1: d C ::/:
where sii2(Xr.,cl)= {fEL00:
II/- Xr.,cl l fXJ < i}.
Let 9' be any arbitrary set which is dense in L 00• Then, for each c with a < c < b, there is a function[cE9' such that
1
II X[8oll]-ft:lloo< 2' since X£•,clEL00 • As suchf., =F [tl for c ::1: d. Hence 9' must be uncountable. This proves that LIXJ is not a separable space-1
6.10 Corollary. No family of continuousfunctions is dense in LIXJ. However, we have the following. 6.11 Theorem.
The family of step functio'!S is dense in L fXJ.
7 BOUNDED LINEAR FUNCfiONALS ON L' SPACES Let p and q be two conjugate exponents. If gE£9 , 1 ~ q ~ oo, it follows from the Riesz-Holder Inequality (cf. Theorem 3.2) that
The Lebesgue L 11 Spaces 221
f· gEL' for each/ELP. As such, for a fixed gELq, one can define a function Fa: I! -+ R by F.<J)= [fg.
Clearly, Fa is a linear functional on the Banach space L'. In fact, we now prove that it. is bounded also. · 7.1 Theorem. Let p and q (I ~ p, q ~ oo) be two conjugate exponents and gELq, Then the linear functional defined by F,(f)=
Jig
is a bounded linear fun,ctional on L11 such that
II Fa II =II g llq.
Proof. First consider the case whenp= oo and q= 1. Observe, by the Riesz-Holder Inequality (cf. Theorem 3.2), that
I F,(f) I ~ ll·g lh II/ l oo •
V /EL'. Thus, it follows that F. is a bonnded linear functional on Y and that '
II Fall
~:11 g
ll1·
To prove the reverse inequality, let• f-sgng. Clearly,/ELCXI and satisfies II/ II ... F.<J) = =>
= 1. Therefore
f fg =J I g I =II g lh
II F. 11=11 g II•·
Let us now consider the case when J < p Riesz-Holder Inequality, ·1 F,(f) I ~ II g llq 11/11,;
<
oo. Again, by the
v/EI!.
Therefore, Fa is a l~near functional on L' and satisfies II F• Further, to obtain the reverse ineq~ty. let
II
~
II g ljq.
I= I g I q-1 sgn g. Clearly, f is a measurable function an41/l'= I g verifies that/EL'. Also, since
1- g=( Ig lq-1 sgn g)·g= Ig lq. 1 if g(x) ~ 0 { •8JDI(x)= -1 if 1/(x) < 0.
IIICq-1>= I g'lq·
This
222 Lebesgue Measure and Integration
we note that
I I I I• -u 1gl·r'(J lg l·t· =(JIll' r''(J I l•t' g
F,(f)-= fg=
g
=>
-11111,11g 11. II F. II ;a. II g 11••
Hence the proof is complete.l We are interested to know ifthe converse of Theorem 7.1 is true in the sense that every bounded linear functional on I! is representable as in Theorem 7.1. For this, we prove the following.
7.Z Theorem (Riesz RepresentatiOD Theorem. Let F be a bounded linear functional on L', 1 :r;;;. p < oo. Then there is a function g in L' such that
J
F(f)= fg,
and that II F II= II g 11•• The proof of Theorem 7.2 needs the following lemma. 7.3 LeDUDB. Let g be an integrable junction on [a, b] and K be a constant such that
1Jfg I :r;;;. Kll/11,. for all bounded measurable functions f. Then gEL1 and II g
111
:r;;;. K.
Proof. First we consider the case when p =1 and q = oo. Let be given, and let E={xE[a, b]: I g(x) I ;;;;, K+E}.
E
>0
Setf=(sgn g)XB. Then/is a bounded measurable function such that
II/ llr• m(E). Therefore Km(E)=KIIfllr;;;;, I
Jig I
=1J g(sgri g)XB I
= JBigl
;;3!: (K+E)m(E)
The Lebesgue L11 Spaces 223 => m(E) = 0, since • > 0 is arbitrary. Hence II g l!oa ~ K. Let us now assume that 1 < p < co. Define a sequence {g.} of bounded measurable functions, where
g.(x)=
i g(x) l0
iflg(x)l ~~~ if I g(x) I > n.
If we get f,.= I g,. I''' sgn g,., then each f,. is a. bounded measurable function such that
IIJ,.II,=( II g,. 11,}11111
I g,. l11 7fn·g,.•J,.·g.
and
Therefore
Jf,.g ~ K II/. 11, =K( II g,. 11 )11111
( II g,. 11,}11 =
11
=>
.(II g,. 11,')11-•111 ~ K (II g,. 1111 ~ K, since q- qfp= I
=>
JI g,. 1 E;;; K•.
=>
But I g,. 111
11
-
I g I• a.e. Thus, by Fatou's Lemma, we have
I Igill~ ~nf f lg,.l ~Kil. 1
Hence, gEL1 and II g Proof of Theorem 7.2. four stages.
11.
~ K.
We shall obtain the proof of this theorem in
Stage 1. Supposef•X1, tE[a, bj, where x, denotes the characteristic function of the interval [a, t]. Set p(t)=F(X,). Clearly, f defines a real-valued function on [a, b]. We first show that f is an absolutely continuous function. Let {]x1, xl[} be any finite collection of non-overlapping subintervals of [a, b] such that I: lxl- x1 I < 8~ If we set. I
then
I= ~(Xx; : xx,) sgn {p(xl)- p(x1}}, ( 11/11,)' < 8,
and so ~ I
I p(x~)- p(x,) I =F{f) Fll· 11/11, II Fll·81''·
~II
<
224 Lebesgue MefJSUre and Integration
Thus 'E I p(x;)- p(x,) I <
E,
for any finite collection of intervals of
total length less than 8 ( .. 11
_:11,) and as such p is absolutely conti-
I
.
nuous on [a, b]. By Theorem VI-7.2, there is an integrable function g on [a, b) such that p(t)•
J:
yte[a, b).
g,
Therefore F(x,)-
Jgx,.
Stage 2. Suppose f is a step function. Since every step function on [a, b) can be expressed as a linear combination of the form 'Ec1x11 with the exception of a finite number of points and F is a linear functional, we have F(f)-
Jgf·
Stage3. Suppose/is a bounded nieasurable function on [a, b]. By Theorem IV-11.1, there is a sequence {1/111} of step functions such that 1/1. -+ f a.e. Since the sequence { I/- 1/1•1"} is uniformly bounded and f- 1/1, -+ 0 a.e., the Bounded Convergence Theorem gives II/-1/1.11,-+ 0 as n -+ oo, and therefore
I F(f)-F(t/1.) 1·1 F(f-1/1.) I
--==fgt/1•.
F(f)= lim F(t/J,)
But, since I gt/1, I ~ 'II g 1. where 'I is the uniform bound of {1/1.}, by Lebesgue Dominated Convergence Theorem, we have
I Hence
fg-
=fgt/1. .
Jfg=F(f), for each bounded :ineasurabte·functionf. Further-
more, since I F(f) I ~ II F in view of Lemma 7.3•.
11·11/11,. we have gEL" and II g 1111
~
II F 11.
The Lebesgue L' Spaces 225 Stage 4. Finally, suppose /EU is any arbitrary function. Let 0 be given. Then, by Theorem 6.4, there is a step function t/J such that II/- t/J II, < •· Since t/J is bounded, we have
•>
F(t/1)= ft/Jg. Therefore
J
I F{f)- fig 1~1 F{f)-F{t/J)+ t/Jg- fig I
~IF(/-t/1) 1+1 f
<
(II
F(f)- ffg. The equality I! F
II= II g 11, follows from Theorem 7.1.1
Problems 1. For p, 1
~
p
<
oo, Prove that
gEL' and I/ I ~ I g I => /EL'· (b) f, gEL', => /gEL'12• If /EL' and gEL, where p, q > 0, then show that fger, for a suitable r. [Hint: Set p' =p'A and q' =q>.. where 1/p + 1/q .. >... (a)
2.
Apply the Riesz-HOider Inequality for functions IJIIJ).EL'' and
I g ll'"eL,'.J
3.
.
Let/ELl such that/is not equivalent to any bounded function.
Show that there is a gELI such tbat/g~£1·. 4. If/is continuous on [a, b], show that lim 11/11,= sup
~
te[a.bJ
lftt) I·
S. Prove Theorem 4.2. 6. Work out the details ~the proof of Theorem 5.2. 7~ Let {/11} be in L', 1 ~ p < oo. Show that if lim
11/.-/11,•0
11-+00
.
Ill
holds in L'; then/11 ~f. 8. Let p and q be conjugate exponents and let/.-+ fin L' and g.~ gin L,. Prove that/.g11 -+ fgin Ll.
226 Lebesgue Measure and Integration
9. Let {/11} be a sequence of functions in L«>. Prove that {/11} converges to f in L «>if and only if there is a set E of measure zero such that {/11} converges to f uniformly on [a, b]- E. 10. Prove Theorems 6.4 and 6.6. 11. Prove Corollary 6.10 [Hint: Apply the method of contradiction and use Theorem 6.9] 12. Let/;;;;:, 0 be such that/EL".p > 0, and let / 11 =min (f, n). Show thatf,EL" and lim 11/-f,.II,=O. 11-+oo
13. A sequence {/11} in L" is said to converge weakly to fin L" if, for every function gEL' with 1/p+ 1/q=1.
Prove that if {/n} converges in the mean of order p to f in L", 1 ~ p ~ oo, it also converges weakly to fin £P. 14. Let fbe a real-valued measurable function on [a, b], 1 ~ p~oo and (1/p)+(1/q)= 1. Then show that
II/ 11, =sup
J:
fg,
where supremum is taken over all real-valued functions g with
II g li1
~
1 and
J:fg < oo.
15. Let {/.} be a sequence of functions in L", 1 < p < oo, which converge a. e. to a function/ in L" and suppose there is a constant Ksuch that 111.11, ~ Kfor all n. Prove that for each function gin L'
f
fg=I~«>f /,g.
Discuss if the result is true for p = 1. 16. Let llfn-/11,-+ 0 as n-+ oo, where{/.} is a sequence in L", andfeLP, (1 ~ p < oo). Suppose {g11} is a sequence of measurable functions such that I g. I ~ M for some constant M, for all n andg,.-+ g a.e. Show that II g./.- gfll,-+ 0 as n-+ oo. 17. A continuousfunctionfi R -+Ris said tohavecompactsopport if there is a compact set Kc:R. such that/(x) =0 whenever x~K. The class of all such functions on R. is denoted by Cc(R.). Further, if E is a measurable set in R., define the class Cc(E) ={/XE: /ECc(R.)}. Prove .that
The Lebesgue L' Spaces 227 (a) Cc(B.)cl!'(R) and Cc(E)cC(E)nLP(E). where C(E) is the class of functions defined and continuous on E. (b) Cc(E)•C(E). if E is compact and, in particular, b] -qa, b];. (c) Cc(E) is dense in L'(E) for 1 :E;;; p < oo.
c.ra.
18. Let C:(R) be the space of hdbdtely dUferentiable faetioas Oh B. with compact support and let C:(E)- {/Xs : /EC:(R)}. E being a measurable set in R. Prove that c;'(E) is dense in LP(E) if 1 :E;;; p <" oo. 19. Prove Theorem 6.11.
Appendix I
Existence of Riemann Integral We give here a criterion for a bounded real-valued function defined on the interval [a, bl to be Riemann integrable therein. Before doing so, we first introduce the rudiments of the Riemann theory of integration. Let/ be a real-valued function defined on a bounded interval I. We define the oscillation offover I, denoted by w(l), as
w(l) = lubf{x)- glbf{x). xel
xel
Definethe oscillation offat eEl, denoted by w(c), as
w(c)=glb w(J), J
where J ranges over an bounded open subintervals of I containing c. Clearly w(J) ~ 0 for any J and as such w(c) ~ 0 for all eEl. Proposition 1. A bounded function/: [a, b] point c if and only if w(c)= 0.
~
R. is continuous at a
Proof. It is easy to verify that the oscillation of f over J is the supremum of the set of numbers I f(xi) -/(X2) 1. where x; and X2 are any two points of J. Let/be continuous at c. Then given an E > 0, there exists a bounded open interval J containing c such that
lftx)-f(c)
I < E/2,
'f/ xEJ.
In particular, if x1 and x2 are any two points of J,
If{x1) -f{x2) I ~ I.tTx1)-.I{a) I+ t.ftx2)-/{a) I <• w(J)
Hence w(c)•O.
<E.
Appendix I 229 On the other band, let w(c)=O. Given an e > 0, there exists a bounded open int~al J containing c such that w{J) < E. This, further, implies If(x1) -/(xi) 1< E, In particular, lf(x)-ftc) I< E, "f/ xEJ Hence I is continuous at c. I Proposition 2. Letfbe a bounded function defined on [a, b). Then f is Riemann integrable if and only if, for each E > 0 there exists a partition p of [a, b) such that S(P)-s(P)
<
E.
Proof. The proof is left to the ~·I
J'roposition 3. If w{x) <,\·for each xE[a, b], then there exists ·a partition P Or [a, b] such that S(P)- a(P)
<
'J.(b- a).
-Proof. For each xE[a, b1 there is a bounded open interval Ja containing x such that w(JJ < .\. Since [a, b] is closed and bounded, by the Beine-Borel Theorem, a finite number of intervals Ja will cover [a, b]. Let P be the set of the end points of these Ja. Let I~o /2, •.. , I,. be the component intervals of P. Then w(I,) < A, for all i= 1, 2, .•. , n. Hence · -It
S(P)- s{P) •
'E ,_, (M1- m,)(x,- x1-1) It
• ,_, 'E w(Ji) (x,- Xt-1) < .\(b- a), I,= ]Xt-t. Xt[ ·I Now the main result is as follows. TheoreJD. A bounded function defined on [a, b) is Riemann integrable if and only if it is continuous almost everywhere. Proof. Let/be Riemann integrable over [a, b] and
E-{xE[a, b): w{x)=O}.
We shall show that m(E)= 0. Observe that
...
E ... U E,, m-1
230 Lebesgue Measure and Integration where
Em={xe[a, b] : w(x)
;;;::
~ }·
It is enough to show that m(.E,.) =0 for each m. Fix m. Since/ is Riemann integrable over [a, b], there exists in view of Proposition 2 a partition P of [a, b] such that E
< 2m ·
S(P)-s(P)
(1)
Thus II
II
~ w(I1)(x1+t-Xt)-= ~ (M,-m,)(Xt+t-Xt)
'-'
t.:'l
=S(P)-s(P) E
<-· ·2m
(2)
Now, E 111 =EmUE:r, where E'm is the set of points of .E,. that are p
points of the partitition P and E:.-E-E:,.. Obviously,
E;.cu J"' k-t
where J1/s are open intervals such that the sum of their lengths does not exceed . E/2. Further, if xeE;:,, then x belongs to some I 1.Hence w(/1)
;;;::
w(x) ;;;::
_!_, m
z,,,
1,2, ••• I,, those of the intervals I, of P each of If we denote by which contains a point of Em, then, in view of (2), the sum of lengths of I 1'JU-1, 2, ... r) does not exceed E/2. Since E' c Ill
.
,
(j J~c
k-t
and
E:.c U I'"' it follows that Em is of measure zero. k-1
To prove the converse, suppose that/ is continuous a.e. in [a, b]. .
ao
.
Since m(.E,.) • 0 for each m, .E,.c U I,, where each 11 is an open
'·-·
interval of (a, b] and* ao
E
lil(I,) < 2w([a, b)j It is clear that .E,. is a closed set in R and hence in [a, b]. Therefore, by the Beine-Borel Theorem, there is a finite number of intervals 111 , 112, ••• , I,, which covers E111• Observe ~t
*We may assume that w([a, bD
>
0.
Appendix I
231
can be expressed as a union of closed intervals Ja, Jz, •••, J,. Thus
[a,
bJ-("~ 1 1," )u(,i; J, )·
Since no interval Jl..t•l, 2, ...,p) contains a point of .E,., there exists, by Proposition 3, a partition P1 of J~ such that S(P1)-s(P1)
<
_l(J,) •
m
Set P•Pa UPzU •.. UP;. Clearly Pis a partition of [a, b]. Hence S(P)- s(P) •·
t is(Pt)- .t(P,)} +- t (Mt~c- m,J (Xt~c- x,,.t) ~-·
t-l
b-a
r
,...
< --+W([a, bD "r...(x,"-x'~c-t)
=•
m
b-a
•
<--,;;- +w([a, bD· 2w([a, b)) < •• by a suitable choice of m for each • is Riemann integrable over [a, b]. I
>
0. Hence, by Proposition 2,
f
Appendhll
Nowhere Differentiable Continuous Functions Wo give here functions which are conunuous on R but nowhere diffe-rentiable. E:umple 1 (Vaa der Waerden). Let F : R. -+ R be a function defined by · F(x) = ~ ftl..x), k-=& 1()k
(1)
where fo(x) =distance from x to the nearest integer ftl..x)=fo(10kx), k=O, 1, 2, ... We observe that:
(a) .Eachf~c is a continuous function on R. •. (b) F is continuous on R.
Since oo f~c(x)
•
1
k~""""f()r <; ~o 2.10"
s
=9<
(xER)
00.
by the Weiersuass M-test, the series (1) converges uniformly on R. Therefore, it follows that F ~ continuous on R because it is the uniform limit of continuous functions. (c) F ia nowhere differentiable, Let aER be any point. It is enough to show that F'(a) does not exist.
Appendix II 233 Suppose a•ao·a1a2t13 •• . a,. ••• For nEN, let X,.""' tzo• a1a:aQ3 , • • a-1b,.a,.+ 1
••• ,
where b,. .. a•+• if a,.r:p 4 or 9 and b,. • a-• if a,= 4 or 9. Therefore
x,.-a-±1o-• lim x,.=a. _.., Thus, for any nEN, we note that fk(x,.)-fk(a)- ±1()k-ll
(k=-0, -1, 2, .. (n-1)
fk(x,.)-fk(a)=O
(k
~
n)
Hence F(x,.)- F(a) • ~- f-,(x,.) -fk(a) x,.-a 10k(x,.-a)
r-o
11-1
-
~
±10k1()1(±10-)
Il-l
- r-o ~- ±1. This verifies that lim [ F(x,.)-F(a) ]
.......
x,.-a
does not exist. Example 2 (Weierstrass). Let F: R -+- R be a function defined by 110
F(x) =,.~
cos 3-x 2,. ,
(xER.).
(2)
Since the series (2) i~ uniformly convergent on R and each term in tho series is a continuous function on R., it foUows that the function F is continuous on R. On the other hand, observe that the series
-,!1, (32 )" sin 3•x, 110
obtained by differentiating the series (2) term by term, is divergent when x is not a multiple of.,., This indicates that F is nowhere differentiable.
234 Lebeague Measure amJ Integration Note. One can construct several examples on the lines of Examples 1 and 2 above. For instance, in Example 1, one may replace the function F by the function defined by
F(x)-~ fk(x),
r-o
,;c
(xEll)
where fG(x) =distance from x to the nearest integer flc(x)=JG(ti'x), k-0, 1, 2, ••• ; while in Example 2, replace, the function F by the function defined by CIO
11,1Jrl cos (a-trx),
F(x) =
where a is an odd natural number and b a real number such that and
ab
371'
> 1+-· 2
At this stage, we may point out that most of the continuous functions are nowhere differentiable. 1his is justified by the Baire category theorem: "The sot 1l is not of the first category.,. (The proof of this theorem is beyond the scope of this book.)
•A set which is the uaion ora denumerable number of nowhere deuse sats is called a set of the lint
_..ry.
Appendix III
The Development of the Notion of the Integral* Gentlemen: Foregoing technical developments, we are going to examine as a whole the successive modifications and enrichments of the notion of the integral and the appearance of other concepts used in recent research concerning functions of real variables. Before c&lichy, there was no definition of the integral in the actual sense of the word "definition". One was limited to saying which areas it was necessary to add or subtract to obtain the integral. For Cauchy a definition was necessary, because with him appeared the concern for rigour which is characteristic of modern mathematics. Cauchy defined continuous functions and the integrals of these functions in nearly the same way as we do now. To arrive at the integral off(~).• it sufficed for him to form the sums (see Figure Ul.l).
S= 'Ef(E,) (XI+I-Xt)
(1)
which surveyors and mathematicians have used for centuries for approximating areas, and to deduce from this the integral
J:f(x)dx by passage to the limit. Although this passage was obviously legitimate to those starting with a notion of area, Cauchy had to prove'that the sumS actually -This is the English translation of Henri Lebesgue's lecture delivered in a coDferenc:e La Societe Mathematique at Copcmhagen on May 8, 1926 and which is included as an appendix in the book Soo Bong Chao, Lebesgue Integration, Marcel Dekker, New York, No. 58 (1980). We wish to acknowledgo our thanks to Marc:c1 Dekker, for permitting us to include the same in this book.
236 Lebesgue Measure and Integration
tended toward a limit under the conditions which he considered. An analogous necessity is imposed each time one replaces an experimental notion with a purely logical definition. It should be added that the interest of the defined object is no longer evident; it can only result from the study of the properties of this object.
0
Fig. m.t.
This is the price of logical progress. What Cauchy did is considerable enough to have a philosophical meaning. It is often said that Descartes reduced geometry to algebra. I would say more readily that, by employing coordinates, he reduced all geometries to that of the straight line and that this geometry, in giving us the notions of continuity and irrational number, has permitted algebra to attain its actual scope. In order that the reduction of all geometries to the geometry of the straight line be achieved, it was necessary to eliminate a certain number of notions related to geometries of higher dimensions such as length of a curve, area of a surface, and volume of a body. Precisely here lies the progress which Cauchy realized. Mter him, it sufficed that arithmeticians construct the linear continuum with the aid of natural numbers to accomplish the arithmetization of the science. And now, should we limit ourselves to doing analysis? No. Indeed, all that we will do can be translated into ~ithmetical language, but if one were to refuse to have direct, geometric, intuitive insights, if one were reduced to pure logic which does not permit a choice among every thing that is exact, one would hardly think of many questions, and certain notions, the majority of those notions which we are going to examine today, for example, would escape us completely. For a long time, certain discontinuous functions were integrated; Cauchy's definition still applied to these integrals, but it was natural to investigate, as Riemann did, the exact scope of this definition.
Appendix III 237
If Ji and / 1 designate the lower and upper bounds of f(x) on (x1, Xt+J), then S lies between and Riemann showed that it suffices that
S- ~= }:.(/,-.{t) (Xt+l- x,) tends toward zero for a particular sequence of partititions of (a, b) into smaller and smaller intervals (x,, x,+l) in order for Cauchy's definition to apply. Darboux added that the usual passages to the limit by ~ and S always give two definite numbers
1:/(x)
dx
J:f(x) dx
These numbers are in general different and are equal only when the Cauchy-Riemann integral exists. From the logical point of view, these are very natural definitions, are they not? Nevertheless, one could say that they are use less in the practical ·sense. Riemann's definition, in particular, has the disadvantage that it applies only rarely and, in a sense, by chance. Indeed, it is evident that partitioning of (a, b) into smaller and smaller intervals (x,, Xt+1) makes the differences J,- Ji smaller and smaller if/(x) is continuous, and by virtue of this contiiiuous process it is clear that this partitioning causes S- S to tend toward zero if there are only a few points of discontinuity. However, there is no reason to hope that the case will be the same for a function discontinue~ everywhere. So, in effect, taking smaller and smaller intervals (x1, x1~1), that is to say, values of f(x) related to values of x which come closer and closer together, in no way guarantees that one takes values off(x) whose differences become less and less. Let us proceed according to the goal to be attained: to gather or group values of f(x) which differ by little. It is clear, then, that we must partiti.on not (a, b), bu~ rather the interval (/,/) bounded by the lower and upper bounds of f(x) on (a, b). We do this with the aid of numbers y 1 differing among themselves by less than e; we are led, for example, to consider values off(x) defined by Yt ~ f(x) ~ Yl+l corresponding values of x form a set E,; in the case of Fig. DI.2, this set E, is made up of four intervals. With a certain continuous function.f{x), it might be formed by an infinite number of intervals. ~
238 Lebesgue Measure and Integration With an arbitrary function, it might be very complicated. But, no matter-it is this set E, which plays the role analogous to that of the interval (x1, Xt+J) in the definition of the integral of continuous functions, since it makes known to us the values of x which give to f(x) values differing very little.
Fig.W.2.
If 'It is any number chosen between y, and Y1+1> Yt ::!i;; 'It ::!i;; Yt+J
the values off(x) for the points of E1 differ from 'It by less than E. The number 'It will play the role which was assumed by f(f1) in (1); as for the role of the length or measure Xt+t- x1 of the interval (x1, x 1_ 1), this will be played by a measure m(E1), which we will assign to the set E1 in a moment. We form in this manner the sum (2) But first let us look at what we have just done and, in order to understand it better, repeat it in other terms. The geometers of the seventeenth century considered the integral of f(x)-the word "integral" had not yet been invented, but that is hardly important-as the sum of an infinity of indivisibles,• each of which is an ordinate, positive or negative, of f(x). Very well! We have simply grouped the indivisibles of comparable size; we have, as one says in algebra, made the collection or reduction of similar terms. It may again be said that, with Riemann's procedure, one 11n the context of areas, indivi8ibles are "infinitely narrow" rectangles of "infinitesimal" area. Leibniz used the symbol dx to denote the "width" of an indivisible, so that the "area" of an indivisible of length y was given by the pro-
duct y dx. He then introduced the symbol
J
y dx for the "sum" or "integral'"
of the areas of the indivisible& which gives the area of a given region.
Appendix III 239 attempted to sum the indivisibles by taking them in the order in which they were furnished by the variation of x. One operated as did a merchant without a method who counted coins and bills randomly in the order in which they fell into his hand, while we operate like the methodical merchant who says I have m(EJ) pennies worth l·m(EJ). I have m(E2) ~ckels worth S · m(E2). I have m(E3) dimes worth lO·m(E3). etc., and thus I have altogether
S= l·m(EI)+S·m(E2)+IO·m(E3)+, •. The two procedures will certainly lead the merchant to the same result because, as rich as he might be, he has only a finite number of bills to count; but for us who have to sum an infinity of indivisibles, the difference between the two methods of adding is capital. Let us now occupy ourselves with the definition of the number m(E,) attached to E 1• The aruilogy between this measUre and a length, or the same with a number of bills, leads us naturally to say that, in the example in Fig. m.2, m(E1) will be the sum of the lengths of the four intervals constituting E, and that, in an example in which E1 is formed from an infinity of intervals, m(E1) will be the sum of the lengths of all these intervals. In the general case, it leads us to proceed as follows: We enclose E1 in a finite or countably infinite number of intervals, and let 1., l2• ..• be the lengths of these intervals. We certainly want m(E,) ~ l1 +l2+ .•• If we look for the greatest lower bound of the second member for all possible systems of intervals which can serve to cover E1, this will be an upper· bound for m(.E,). For this reason we ·denote it by m(Ei) and we have
m(E1)
If Cis the set of similarly
~ints
~
m(.E,)
of (a, b)
n~t included in
(3) E1, we have
m(C) ~ m(C) Now, we obviously wish to have m(E1)+m(C)=m((a, b))=b-a Therefore we must have m(E1) ~ b-a-m(C)
(4)
240 Lebesgue Measure and Integration
The inequalities (3) and (4) give then the upper and lower bounds of m(E,). One can easily see that these two inequalities are never contradictory. When the lower and upper bounds of E1 are equal. m(E;) is
defined, and we say then that E 1 is measurable.2 A functionj(x) for which the sets E, are measurable for all y 1 is called measurable. For such a function, formula (2) defines a sum S. One can easily prove that, when one varies the choice of y1 in such a way that " tends toward zero, the sum S approaches a definite limit which is by definition
J:f(x) dx.s This first extension of the notion of integral led to many others. Let us suppose that it is a question of integrating a functionf(x, y) in two variables. We proceed exactly as before. We assign to it sets E, which are now points in the plane and no longer the points on a line. To these sets we must now attribute a plane measure; this measure is deduced from the area of the rectangles entirely in the same manner as linear measure is deduced from the length of intervals. With the measure defined, formula (2) will give the sum S, from which the integral is deduced by passage to the limit. The definition which we have considered thus extends itself immediately to functions of several variables. Here is another extension which applies equally well whatever the number of variables, but which I state only in the case where it is a question of integrating f(x) on (a, b). I have said that it is a matter of forming the sum of indivisibles represented by the various ordinates of the points x, y=f(x). A moment ago, we grouped these indivisibles according to their size. Let l'fhe method of defining the measure of sets used here is that of C. Jordan (Cours d'Analyse de rEcole Poly technique, Vol. 1) but with this modification essential to our aim; that we enclose the set Ei to be measured in intervals which may be infinite in number, whereas Jordan always used only a finite number of intervals. This use of a countable infinity in place of integers is suggested by the endeavors of Borel who, moreover, himself used this idea in particular for a definition of measure (Le~ons sur Ia thlorie us /one lions). 'Comptes Ren:dus Acad. Sci. Paris, 129, 1909. Definitions equivalent to that of the text were proposed by various authors. The most interesting are due to W.H. Young (Philos. Trans. Roy. Soc. London, 204, 190S, Proc, London Math, Soc, 1910). See also, for example, the notes by Borel and by F, Riesz (Comptes Ren:dus Acad. Scl. Paris, 154, 1912),
Appendix III 241
us now restrict ourselves to grouping them according to their sign; we will have to consider the plane set E11 of those points, the ordinates of which are positive, and the set E, of points with negative ordinates. For the simple case in which/(x )is continuous, before Cauchy, as I recalled in the beginning, one wrote J:/(x}dx-area (E,)-area (E,) This leads us to write
J:
/(x} dx= m.(E,}- m.(E,)
m. desipting a plane measure. This new definition is equivalent to the preceding one. It brings us back to the intuitive method before Cauchy, but the definition of measure has given it a solid logical foundation. We thus know two ways of defining the integral of a function of one or more variables, and tlui.t we know without having to consider the more or less complicated form of the domain of integration, because the domain D intervenes only as follows: The sets E 1 of our first definition and the sets E11, E, of the second are formed by taking values of the functionfonly on the points of D. Since the choice of the domain of integration D enters only in the formation of the sets E1, or E, and E,, it is clear that we could just as well agree to form these sets E1, E,, E, by taking into consideration only the vallle$ assumed byf on the points of a given set E, and we wiQ have hence defined the integral off extended to the set E. In order to make precise the scope of this new extension of the notion of integral, let us recall that our definitions require that/ be measurable, that is to say, that the sets E1 be measurable for the first definition, and that E11 and E, be measurable for the second, and, in view of this, E must also be measurable. We thus know how to define the integral extended to a measurable set of a measurable and bounded function on this set. I have, in effect, implicitly supPosed thus far that we are dealing with bounded functions. What would have to be changed in the first manner of definition if the function f to be integrated were not bounded? The interval (f, J) would no longer be finite; an infinity of numbers Yt would be needed to divide it into intervals of length at most equal to E, so there would be an infinity of sets E1 and the sum S of formula (2) would now be a series. In or~r not to be stopped at the outset, we must assume that the series S is convergent for the first choice of the numbers Yt
242 Lebesgue Measure and Integration that we would make; but, if S exists for one choice of y, it exists for all choices of y 1, and the definition of the integral applies without modification. The name of summable functions has been given to all functions which can be integrated by the indicated procedures, that is to say, to all measurable functions for which the sums S have a meaning. Every bounded measurable function is summable; and as no one has up to now succeeded in naming a nonmeasurable function, one could say that, up to now, practically every bounded function bas an integral. On the contrary, there exists very simple unbounded functions which are not summable. Thus, one must not be astonished that our notion of integral still reveals itself insufficient in certain questions. We have just extended the notion of integral to unbounded functions by starting with the first of our definitions; the second leads to the same result. But for this it is necessary to enlarge the notion of measure in such a way that it applies not only to bounded sets, which we thus far considered solely, but also to sets of points extending to infinity. I mention this second method of proceeding only because it is also related to another extension of the definite integral in which the interval, the domain, the set on which the integral is extended, is no longer presumed finite, as we have done up to now, but may go to infinity. . I limit myself to just an indication, because I will not be considering "in what follows this extension of the integral concept. It is for the same reason that I am content with mentioning briefly the research, still very original, undertaken by a young man killed in the war, R. Gateaux, who intended to define the operation of integration for functions of infinitely many variations. This research, which was continued by Paul Levy and by Norbert Wiener, is not without relation to the axiomatic studies undertaken by M. Frecbet and by P.J. Daniell with the aim of extending the notion of integral to abstract sets.4 Frechet and Daniell proposed furthermore to apply to abstract sets not only the definitions of which I have spoken thus far, but also a further extension of the definite integral, to which we shall be led soon by the notion of indefinite integral, which we are now going to examine.
'R. Gateaux, Bull. Soc. Math. France,1919; P. Uvy, Lecons d'analyse /onctionelle, 1922; N. Wiener, Proc. London Math. Soc., 1922; M. Fr6chet, Bull. Soc. .Math. France, 191S; P.J. Daniell, Ann. of Math., 1918 and 1919.
Appendix Ill 243 On.e ordinarily calls the indefinite integral of a function /(x) the function F(x) defined by F(x)=C+
J:/(x)dx
(5)
We do not adhere to this name but give rather to the words "indefinite integral" their original meaning. OriginaUy, the two names "definite integral" and "indefinite integral" applied to the same expression
J:
f{x) dx. But the integral was called "definite" when it was a
question of a given, determined; or defined interval (a, b); and the integral was "indefinite" when (a, b) was variable, undetermined. undefined, qr, if one wishes, indefinite. It is, in short, by a veritable abuse of language that one calls F(x) the indefinite integral of f(x). If we remark in addition that when one studies F(x) it is always to obtain properties of it is actually
J:
J:
f(x) dx, that
f(x) dx which one studies through F{x), one will be led
to say: I call the indefinite integral off(x) the function f'(a, b)= J:f(x)dx=F(b)-F(a)
(5')
There are between an indefinite integral and the corresponding definite integral the same relations and same differences as between a function and a particular value taken on by this function. Furthermore, if we represent by D the interval (a, b) of integration, we may say tha~ the indefinite integral is a function, the argument of which is the dom._ain D, 1/J(D)= f'(a, b)
From these reflections it clearly results that, relative to a function of two variablesf(x, y), one must not take for the indefinite integral, as is sometimes done, the function F(;, Y) = CJ(X) + c2(y) +
J: J:
f(x, y) dx dy
{6)
If one limits oneself to considering rectangular domains fJ :o::;;; X :o::;;; b, C :o::;;; y :o::;;; d one must take for the indefinite integral the function of four variables ,(a, b; c, d),=F(b, d)+F(a, c)-F(a, d)-F(b, c)
(7)
244 Lebesgue Measure and Integration But if one wishes to oonsider all the domains of integration, since the most general domain cannot be determined by a finite number of parameters, however large the number, it becomes necessary to give up ordinary functions to represent the correspondence between a domain D and the integral .extended to this domain and to study directly the function r/J(D)=
ff
0
f(x,y)dxdy
for which the argument D is a domain. It is this function which we will call the indefinite integral of f(x, y). Or rather, since we have also defined the integral off extended to a measurable set E, we will consider the indefinite integral as a set function which will have been defined for all measurable sets. 5 In all that has been said up to now, there are, to be sure, only questions of language or of naming; but those questions would not have been asked if we had not acquired a new concept. It is for this reason that one should not be surprised that the new language has allowed one to give all possible meaning to facts perceived first of all in the case of the function F(x) of formula ( 5). One has succeeded, in particular, in characterizing set functions which are indefinite integrals by two properties: complete additively and absolute continuity.' When a set function possesses these two properties, it is the indefinite integral of a function/which depends on 1, 2, 3, ... variables according to whether the sets E are formed with the aid of the points on a line, in a plane, in ordinary space, etc. In order to have a uniform language and notation, let us say that/ is a point function,f(P); We write
f
r/l(E) =
E
f(P) dm(E)
(8)
The function/(P) is entirely determined by 1/J(E) to the extent that one can arbitrarily modify f on the points of an arbitrary set of measure zero without its ceasing to have 1/J(E) for an indefinite integral. And one can obtainf(P) starting with 1/J(E), except on points of a set of measure zero, by the following procedure. Ann. Sci. del' Ecole Normale Suplrieure, 1910. •These terms are due respectively to de Ia Vallee-Poussin (lntegrales de Lebesgue, fonctions d'ensemble, c/asse! de Baire, Paris, 1916} and G. Vitali (R. Ace. Sci. Torino, 1908). A function of a measurable set is absolutely continuous if, when E varies in such a way that m(E) tends toward zero, r/t(E) also tends toward zero. "Complete additivity" is a synonym for '"countable additivity". 1
.Appendix III 245 Let P be the point at which we wish to calculate f; we take for the domain of integration L1 an interval with center P, or a circle with center P, or a sphere with centre P-according to whether we are dealing with the case of the line, plane, or space-and we form the ratio 1/J(.LI)fm(LI). Then, let Ll tend to zero and we have lim 1/J(LI) =f(P) · (9) A-+D m(LI) This result evidently generalizes the classical theorem according to which, f(x) is continuous, the function F(x) of formula (5) admits f as its derivative; our procedure of calculatingf(P) is indeed, in effect, a sort of differentiation of the set function 1/J(E). This manner of differentiation was considered quite a long lilne ago. Cauchy7 calls ''coexistent quantities" those quantities determined at the same time, that is, by the same conditions. If, for example, one has a nonhomogeneous body, nonhomogeneous in composition and density, and if one, considers a domain.D of this body, the volume of D. the mass of D, the quanpty of heat necessary to elevate by one degree the temperature of' D supposed isolated, all are coexistent quantities. These are functions V(D), M(D), Q(D) of the domain. It is n·ot by happy chance that we arrive here at functions of domains. lf one reflects on it, one quickly sees that every magnitude of physics is related not to a point, but to an extended body, that it is a function of a domain, at least insofar that it is a matter of directly measurable magnitudes. The body to be considered will not, however, always be a body of our customary space; it could be a body in a purely mathematically conceived space if, in the determination of.the envisaged magnitude, there intervene nonspatial variables such· as tiriie, temperature, etc. But this is of little importance; directly measurable magnitude-mass, quantity of heat, quantity of electricity, for example-are functions of· a domain and not functions of a point. Physics meanwhile also considers magnitudes associated with points, such as speed, tension, density, specific heat; but these are derived magnitudes which one defines accurately most often by the ratio or the limit of the ratio of two coexistent quantities: D
"t Mass ensl Y= Volume ;
S ifi h t Quantity of heat pee c ea = Mass ,
that is to say, tiy(takingthe derivative of a magnitude with respect to a coexistent quantity. 'Exercices d'analyse etde physique mathemat/que, Vol. 2, Paris,l840-1847, pp. 188-229.
246 Lebesgue Measure and Integration Thus physics, and consequently geometry, leads to the consideration of functions of a domain and their differentiation just as does analysis of functions of real variables. Similarly the functions of a domain have, in physics, a somewhat more primordial role than point functions. Why then do physicists not speak of these functions? Because mathematicians have not yet studied them and because algebra has notation neither for the domains, nor for the functions of domains. Thus one sees the physicist limit himselfto considering special domains depending only on ~ertain parameters, in such a way that the domain function to be considered is reduced to a function of parameters. This is, moreover, exactly what a mathematician does when, instead of considering the definite integral of f(x, y) in all of its generaiity, he limits himself to considering the functions F(X, Y), p(a, b; c, d) of formulas (6) and (7). We remark furthermore that formula (8) establishes a connection between the set functions rfl(E), which are indefinite integrals, and point functionsf(P), which are dependent uponalgebra. This formula (8) thus furnishes a sort of notation for certain set functions. But when one examines the two conditions required for a function to be an indefinite integral, one cannot doubt that physical quantities are among the class of functions susceptible to the notation. These reflections on the nature of physical quantities may have allowed you to understand more precisely the interest and the importance of the notions which we have encountered. They show, in particular, that the operation of differentiation which appears in formula (9) is not the only one to be considered, that one can always consider the differentation of a function r/J(E) with respect to a coexistent function p(E), whether or not it is the measure m(E). One question now quickly comes to mind: Can one also replace the function mlE) with a given functionp(E) in the definition ofthe intergral? In this there is no difficulty. We will first replace formula (2) . by
S= l).£,p(E,) if first the sets E1 belong to the family of those sets for which the function p(E) is defined, that is, the function to be integrated must be measurable with respect to p(E) in order for the series s to be convergent, that is,fmust be summable with respect to p(E). This being presumed, the definition of the integral off(p) with respect to p(E), If(P)dp(E)
Appendix Ill 247 is obtained as before if the function p(E) and possesses a certain property which one expresses in saying that p(E) must be of bounded variation.• We have just arrived at a new and very considerable extension of the notion of integral in taking the formal point of view of the mathematician; the point of view of the physicist leads even more naturally to the same result, at least for continuous functions f(P). One could similarly say tlult the physicists have always considered only integrations with respect to domain functions. Suppose, for example, that one wishes to calculate the quantity of heat p(D) necessary to elevate by one degree the temperature of a body D of which we spoke above. One must divideD into small partial bodies D., D 2, •.• of masses M(D1), M(D,), ... , choose from each a point P 1, P2, ••• and choose for an app-:oximate value of p(D) the sum /(PJ)M(D•)+f(P2)M(D2) +. . . . f(P) designating the specific heat at P. This is to say that we are calculating p(D) by the formula . p(D) =
J D
f(P) dM(E).
In its general form the new integral was defined only in 1913 by Radon; it was, meanwhile, known since 1894 for the particular case of a continuous function of a single variable. But its first inventor, Stieltjes, was led to it by research in analysis and arithmetic, and he presented it in a purely analytical form which masked its physical significance so much that it required much effort to understand and 8p(E)"is said to be of bounded variation if, in whichever manner one partitions E into a countably infinite number of pairwise disjoint sets E1o E 1, • • • , the series E I p(E,) I is convergent. The notion of functions of bounded variation was first introduced by C. Jordan for functions of one variable. The only set functions p(E) to be considered in these theories are additive functions, that is, those for which one has p(& +Ea+ • . •)=p(EJ+p(Ea)+ . •• E1 , E 1, • • • being pairwise disjoint. If the additivity is complete, that is, if the sequence &. E 1 , ••• can be chosen arbitrarily, p(E) is necessarily of bounded variation. In eft"ect, the order of the sets be~ unimportant, · the series p(&)+p(E1)+ ... must remain convergent whatevet the order; that is, the series J I p(E,) lis convergent. No attempts have been made up to now to get rid of the condition that p(E) be of bounded variation. One ought to remark besides that if p(E) were not of bounded variation, one could find a continuous function/(p) for which, neverthelses, our definition of intepoai would not apply.
248 Lebesgue Measure and Integration
recognize what is now evident. The history of these efforts cites the names of F. Riesz, H. Lebesgue, W.H. Young, M. Frechet, C. de Ia Valtee-Poussin; it shows that we competed in ingenuity and in perspicacity, but also in blindness.~' And yet, mathematicians always considered Stieltjes-Radon in-
J
tegrals. The curvilinear integral c f(x, y) dx is one of these integrals, relative to a function defined in terms of the length of the projection onto the x axis of a arcs of C; the integral J J 8 f(x, y, z) dx dy involves in the same way a set function defined in terms of areas of S projected onto the xy plane. In truth, these integrals most often present themselves in groups Jcf(x, y)dx+g(x, y)dy
Is
f(x, y, z)dxdy+g(x, y, z)dydz+h(x, y, z)dz dx
If one thinks also of integrals considered for the definition of lengths of curves or areas of surfaces, Jc (dx2+dy2+dz2) 1t2
Jt
[(dxdy)2+(dydz)2+(dzdx)2]112
one will be led to say that it is also convenient to study modes of integration in which there appear several set functions Pt(E), p2(E), . . . . This study remains entirely for the future, although Hellinger and Toeplitz have utilized certain summations with respect to several set functions.to We have thus for considered integration, definite or indefinite, as an operation furnishing a number, defined or variable, by a sort of generalized addition. We are placed with the point of view of quadratures. But one may also consider the integration of a continuous function as furnishing a function, just like the most simple of integrations of differential equations. It is this point of view of primitive functions which we will now consider. Finding the primitive function F{x) of a given function f(x) is finding the function, determined to an additive constant, when it exists, •1. Radon, Sitz, Kals. Ale. Wiss. YieiUUI, Vol. 122, Section Ila, 1913; T.J: Stieltjes, Ann. Fac. Sci. Toulouse, 1894; F. Riesz, Comptes Rendu Acod. Sci. Paris, 1909; H. Lebesgue, ibid., 1910; W.H. Young, Proc. London Math. Soc., 1913; M. Frecbet, Noll'l. Ann. us Math., 1909; de Ia VaiiCe-Poussin, op. cit. 10Sec, for example, J. reine angew. Math., 144, 1914, pp. 212-238.
Appendix Ill 249 which admits f(x) as its derivative. It is this problem that we are going to study. But first we remark that the preceding re1lections lead to formulating the problem in a much more general fashion: Given a function f(P) which is the derivative with respect to a known function p(E) of an unknown function 1/J(E), find the primitive function 1/l(E) off(P). If, for example, we are dealing with a continuous function /(x) and if m(E) is the m~asure, the primitive function would no longer be the functipn F(x)
o~.formula(S), buttheindefiniteintegral J8 f(x) dx.
I can only mention this general problem which has not been studied; I ~ content with remarking that the Stieltjes integral would be very insufficient for resolving it. This integral has, in effect, only been defined for the hypothesis that p(E) is of bounded variation, and one may certainly speak of differentiation with respect to a ·function p(E) with is not of bounded variation. The theory of summable functions furnishes the following result related to the case in which p(E) is the measure m(E): When the derivative f(P) is summable, the antiderivative off is one of its primitive functions. I say one of its primitive functions because one still does not know very well now this general problem of primitive functions must be posed in order for it to be determined.JI Let us leave aside thses questions, which I speak of only in order to show how much there remains to be done, and let us show how much has been done in the search for the primitive function F(x) off(x), thanks above all to Arnaud Denjoy. · I have just said that, whenf(x) is summable, integration furnishes F{x) by formula (5). Suppose that, on (a, b),f(x) fails to be summable only at a single point c. Then integration gives us F{x) on (a, c-E) for arbitrarily small E and hence on ·the whole interval (a, c); it also gives F(x) on (c+E, b) and hence completely on (c, b). And taking into account the continuity of F(x) at the point c, we have F(x) on the whole interval (a, b). By such considerations of continuity, 12 one sees that, if one knows F(x) on every interval which contains no point of a set E in its interior or at its extremeties, one can deduce F(x) by an operation which I shall designate by A on every interval adjacent to E, that is, on every interval having its end points in E but- havipg no points of E in its interior. nsee on this subject the notes of Fubini and Vjtali, appearing 191S-1916, in A. ttl Rend. R. A.cc. Lincei. · · 11lt is the introduction of these conditions of continuity which very considerably
difFerentiates the problem of primitive functions from that of quadratures.
250 Lebesuge Measure and lnteration Suppose now that one knows F(x) on intervals(«, fJ) adjacent to a set E, that the sum l:[F(P)-F(«)] is convergent, and that/(x) is summable on E.l 3 Then it suffices to say that the primitive function must result from the contribution of E and the intervals adjacent toE in order to be led to the formula
F(x)-F(a)=~E
/(dx+l:(F(p)-F(ct)J}:
the braces of the second member indicating that one must utilize there only points between a and x. From this formula there results the determination of F(x), thanks to an operation with I will designate by B. The preceding results mark the extreme points which I reached in my thesis, and I must say that I indicated them only somewhat by chance, because I did not at all suspect the importance given to them by Denjoy. Relying on Baire's results, Denjoy shows that, if f(x) is a derivative function on (a, b), then (1) The points for whichf(x) is not summable form a set E1 which is not dense in (a, b); an operation 01 of type A. determine F(x) on intervals adjacent to E 1• (2) Next, there exists a set & formed from point of E 1 and not dense in E., on the adjacent intervals of which one can calculate F(x) by an operation 02 of type B. (3) Next, there exists a set E3 formed from points of E2 and not dense in &, on the adjacent intervals of which one can calculate F{x) by an operation 03 of type B, .... If it turns out that after an infinite sequence of operation o., 02, ... , one has not yet found F(x) on the entire interval (a, b), the points of (a, b) which are not interjor-points of intervals on which one has defined F(x) form a set E.,, and an operation of type A., the operation 0.,, furnishes F on intervals adjacent to E.,. One considers next, if it is necessary, operations 0.,+., 0 .. +2•••• of type B, followed by an operation 02.. of type A., followed by operations of type B, etc. And Denjoy, using now classical arguments of Cantor and Bendixson, proves that this procedure will finally give us F{x) on the entire interval (a, b) after a finite or countably infinite number of operations. 11It is convenient to remark that these hypotheses are not contradictory, the same as if E is assumed to be the set of points on which/(x) is not summable in an interval (a, b) considered. For the determination of points of nonsummability on (a, b) it is necessary, in effect, to take into account all points of (a, b), whether they belong to E or not; whereas summability on E is a condition occurring only on the points of E.
Appendix III 251 This operative procedure, certainly complicated, but just as natural, in principal, as those previously envisaged, was called by Denjoy "totalization". Totalization solves entirely the problem of finding the primitive function F(x) of a given functionf(x); it permits at the same time the determination of F(x) knowing only a derived number14 f of F(x) and no longer its derivative. I shall not dwell on these beautiful results; the most important fact for us is that totalization, by a long detour, flll'lli;shes us ·with a new extension of the concept of definite integral. Every time, in effect, that totalization applies to a function f(x) and gives a corresponding function F(x), we can attach to f(x) an integral, tllanks to formulas (5) and (5').15 Gentlmen, I end now and thank you for your courteous attention; but a word of conclusion is necessary. This is, if you will, that a generalization made not for the vain pleasure of generalizing, but rather for the solution of problems previously posed, is always a fruitful generalization. The diverse applications which have already taken ~e concepts which we have just examined prove this superabundantly.
11Dini's derivative. , U'fhe detailed memoirs of Delijoy appeared from 1915 to 1917 in the Journal de Math., in the Bull. Soc. Math. France, and in the Ann. Sci: de l'E'cole Nol'lnQle Superleure.
·Bibliography 1. Barr~ G .De. 'Introduction to Measure Theory', Van Nostrand Reinholcl.Company, New York (1974).
2.
3.
4. S. 6. _7.
8. 9. 10. 11. ·
12. 13. 14.
15. 16.
B~rrill,
C.W. and Knudsen, J.R. 'Real Variables', Holt Rinehart and Winston, Inc.~ New York (1969). Chae, Soo Bang. 'Lebesgue lnteration', Monographs and Textbooks in Pure and Applied Mathematics No.- 58, Marcel Dekker, Inc., New York (1980). __ _Goldberg, R.R., 'Methods of Real Analysis', Oxford & IBH Publishing Company, New Delhi (1973). Haaser, N.D. and Sullivan, J.A. 'Real Analysis', Van Nostrand, Reinhold Company, New York (1971). ·Halmos, P.R. 'Measure Theory', Van Nostrand, Princeton, New Jersey (1950). Hu, S.T. 'Elements of Real Analysis', Holden-Day, Inc., San Francisco (1967). McShane, E.J. and Botts, T.A. 'Real Analysis', D. Van Nostro. and Co. Princeton, New Jersey (1959). · Mukherjea, A. and Potboven, K. 'Real and Functional Analy-· Sis', Plenum Press, New York (1978). Natanson, I.P: 'Theory of Functions of a Real Variable Vol I, Frederik Ungar Pub. Company, New York (1965). Randolph, J.F. 'Basic Real and Abstract Analysis', Academic Press, New York (1968). Royden, H.L. 'Real Analysis (2nd ed.)', The Macmillan Company, New York (1968). Saxcma, S.C., and Shah, S.M. 'Introduction to Real Variable Theory', Printice-Hall of India Private Limited, New York (1980). Spiegal, M.R. 'Schaum's Outline of Theory and Problems of Real Variables', Schaum's Outline Sories, McGraw-Hill Book Company, New York (1969). Taylor, A.E. 'General Theory of Functions and Integration', Blaisdell Pub. Company, New York (1965). Wheeden, R.L., and Zygmund, A. 'Measure and Integral', Marcel Dekker, Inc., NC\V York (1977).
List of Symbols and ~Notations E
~
c
::::>
u n ~
3
I
v
N I
Q R. [+
N-
Q.+ QR..
R.•
A-B AxB A.AB A...wB
AlfiB A,c
A,B
A' int (A.) EP(A.)
""' .
belongs to does not belong to is a subset of, is contained in contains miion intersection empty set there exists Q.E.D. for all set of natural numbers set of integers set of rational numbers set of real numbers, real number system· set of nonnegative integers set of negative integers set of positive rational numbers set of negative rational numbers set of extended real numbers, extended real number system . . n-dimensional Euclidean space the set of all points in A which are not in B cartesian product of A and B symmetric difference of A and B A is equivalent to B A is not equivalent to .B the complement of A the set of functions on B with range in A the derived set of A interior of A power set of .A, set of subsets of. A characteristic function of A
List of Symbols and Notations 25J c(A) or
A
1 or card (A) Ko c
c C(«)
q,}
g.
~
!B
f: A-+ B J-•: B-+ A /(A) /t.A
gof: A-+ B J-l(b) J-•(B)
,_
f+ -
, .....g f=g
[x] 0
x+y ·xRy 0
A+y I(I) m(E) in~(E)
m.(E) D+j'(x) D+f<x) D-f(x) D-f(x) BV[a, b] Vf
T!(f) C[a, b]
IHI (X,II·II) LP(E) L'1E)
closure of A cardinal number of A aleph nought cardinal number of the continuum cantor set generalized cantor set special Borel sets family of measurable sets class of Borel sets function (or mapping) with domain A and range in B inverse function (or mapping) image of A under the function/ the restriction off to A composition of mappings inverse image of b . inverse image of B positive part of the function/ negative part of the function/ f is equivalent to g f is equal to g greatest integer not greater than x sum Dio4ulo 1 of x and y x is relat~d to y translate modulo 1 of A by y leilgth of the interval I measure of E outer measure of E inner measure of E upper right derivative lower right derivative upper left derivative lower left derivative space of the fun~ons of bounded variation on [a, b] variation function total variation off on [a, b] space of continuous functions on [a, b] norm normed space X class of p-integrable functions over. E · class of measurable essentially bounded functions onE
Index Absolutely continuous function, 192 convergent sequence, 11 summable sequence, 202 summable series, Accumulation point, 9 Addition of cardinal numbers, 37 Adherent point, 7 Aggregate, l Algebra of sets, 66 Algebraic number, 26 Almost everywhere (a.e.),lOS uniform convergence, 112 Asymptotic convergence, 118 Axiom of choice, 20 Baire category theorem, 234 Banachspace,200,201,213 Boolean algebra, 66 cr-algebra, 70 Borel field, 70 function, 108 measurable function, 108 set, 72
Bounded above,S below, S convergence theorem, 142 linear functional, 204, 220 setS, variation function, 179 Canonical representation, 102, 130 Cantor continuum problem, S1 . function, Y
like sets, 44 n-8ry sets, SO
set, 44 ternary set, 44 theorem,36 Cardinal number, 33 Cartesian product, 2, 4 Cauchy inequality, 211 Schwarz inequality, 211 sequence,!) sequence in measure, 123 Characteristic function, S1, 100 Choice function, 4 Closed at left, 6 interval, 6 set, 7 Closure, 7 Codomain,3 Collection, 1 Compact support, 226 Complement, 2 Complete ~rmed space, 201 Composite function, 4 Conditionally continuous function, 77 Conjugate exponents (numbers), 208 Constant function, 4 Continuous function, 12, 103, 112, 113 Continuum, 30 hypothesis, so Convergence absolute, 11 almost everywhere, 9 almost uniformly, in the mean, 216 in the mean of order p, 216 in m~ure. 111, 118
Index 257 in the norm, 216 pointwise, 11 uniform, II Convergent ·sequence , 9, 78 series, 10 Countable additivity, 54 set,19 subadditivity, 58 Countably infinite set, 20
eove.-. s
Decreasing sequence, 9 Dense set, 9 Denumerable set, 19 Derivatives, 13 Derived set, 9 DifFerence, 2 DifFerentiation, of an integral, 185 or monotone functions, 173 D'ini Derivatives, 169,170 Disjoint, 2 Domain, 3 Egorofl"s theorem, 113 Elements, I Elementary intogra1, 128 Empty set, 2 Enumerable set,'19 Enumeration, 20 . Bquimeasurable set, 74 Equivalence classes, 5 relation, S Equivalent functions, 213 sets, IS Essential bound, 20S supremum (ess sup), 107,.205 . infimum (ess inf), 107 Essentially bounded function, 205 Exponentiation of cardinal num~ 40 Extended real number of system, 6 real valued function, 89 Bxtonsion or a function, 146
Fatou's lemma,l46 (-image, 3 Finite cardinal number, 3S cover, 8 set, 19 F. Riesz theorem, 121 First category, 234 Frechet theorem, 117
q,.set,60 9'..a-set, 61 9'o8o•set, 61
Function, 3 absolutely continuous, 192 Borel,108 bounded variation, 179, 180 Cantor, SO cba~stic,51, 100 choice, 4 composite, 4 continuous, 12, 103, 112, 113 decreasing, 14 derivable, 13 difFerentiable, 13 essentially bounded, 20S identity, 4 increasing, 14 indicator, 100 infinitely difFerentiable, 227 integrable, 150 measurable, 88, 89 monotone, 14 negatiw variation, 183 nowhere difFerentiable, 169, 232 positive variation, 183 simple,101 step, 94 strictly decreasing. 14 strictly increasing, 14 variation, 182 Fundamental sequence in measure, 123 Fundamental theorem or calculus, 169 Generalised cantor set, SO continuum hypothesis, Sl ga ·set, 61 gao-set, 61 g.,.a-set, 61
258 Index Greatest lower bound (glb), 61 Beine-Borel theorem, 8 Holder's inequality, 208 Identity function, 4 Improper integrals, 1,65 Increasing sequence,· 9 Indefinite integral, 185 lndex,3 set, 3 Indexed set, 3 Indicator function, 100 Infimum (inf), 6 Infinite set, 19 Infinitely diflerentiable function, 227 Inner (interior) measure, 64 Integrable function, 133, ISO Integral of the derivative, 196 Integration by parts, 198 Interior point, 7 set, 7 Intersection of sets, 2 Intarval, 6 Isolated set, 27 Jordan decomposition theorem, 182
interval, 53 open set, 54 sets, 53 Limit inferior (lower), 10, 78 point, 9 superior (upper), 10, 78 Linear functional, 204 · Lipschitz condition, 196 Lower bound, s left derivative, 170 limit,10 right..cferivative,170 L""-spaces, 206 L~-spaces, 205 , ....spaces: 202 1~-spaces, 201 Lusin' theorem, 115 Measmable function, 88, 89 set, 64 Measure, 6S Members,! Metric induced by norm, 201 Minkowski's inequality, 208 · Monotone convergence theorem, 147 function, 14 sequence, 9 Monotonicity, 56 Multiplication of cardinal numbers, 38
Least upper bound (lub), S Lebesgue dominated con~ce theorem, 160 ioner measure, 64 integrable function,133 integra1,127, 130,134,144,154 lower integral, 132 Nearly uniform convergence, 216 measure, 65 Negative measure function, 6S measurable function, 89 part of a function, 98 variation of a function, 183 measurable set, 64 outer measure, SS Neighbourhood (nbd), 7 point,191 . Nested interval property, 29 sot, 190,191 Non Borel measurable upper integral, 132 function, 108 Lebesgue's theorem, 176 set, 74 Left hand Nondense set, 9 derivative, 14 Nondenumerable set, 28 limit, 12 Nonmeasurable function, 101 I.ensthof closed set, S4 set, 83
Index 259 Norm, 200, 204 Normed linear space, 200 space, 200 Nowhere dense set, 9 Nowhoni differentiable continuous function, 16~, 232 Null set, 2 Qno.one,3
Ono-to-One, 4 Onto, 3
Open
at right, 6 cover, 8 interval, 6
set,7 Operation on sets, 2 Oscillation, 228, Outer measure, SS Pairwise disjoint, 3 Partition, S Perfect set, 48 p-mean Cauchy sequence, 216 Point, I of closure, 7 Pointwise convergenee, ~1 Positive part of a function, 98 variation of a function, 183 Power set, 2 Pre-image, 3 Presque parpout, 106 Product, 2 Proper subset, 2 Quotient set, S
Range.3 Rational point, 27 Real valued function, 3 Reftexiw, Relation, 5 Restriction of a function, 4 Riemann intesraJ, 128 Riesz theorem, 121
s
Fiaher theorem, 214
Holder inequality 208, 212 Minkowski's inequality, 210, 212 representation theorem, 222 Right-hand derivative, 14 limit,12 Schroder-Bernstein, theorem, 18 Semi closed interval, 6 open interval, 6 Separable space, 220 Sequence, 9 Cauchy,9 decreasina,9 increasins, 9 monotone, 9 Series, 9, 10 Set,1 Borel, 72 closed, 7 dense, 9
q, ,60
!la,61 isolated, 27 Lebessue.190, 191 measurable, 64 non-measurable, 83 nowhere dense, 9
open,7 perfect, 48 Set function, 53 Set of first catOJOl)', 234
measure zero, 105 Ssnx, 221 a-algebra, 70 Simple function, 101 Singular function, 196 Step function, 94 Strictly decreasin& 14 increasins. 14 Subcover,8 Subset, 2 Subsequence, 10 Subspace,194 Sum,2 modulo 1,83
$ammablo
R60 Index function, 150 sequence, 202 Super set, 2 Supermum, S Symmetric,
s
dift'ennco, 66 Total volume of a function, 180
Tranacendental Dumber, 26 Transfli1ite carcfiDai Dumber, 35 Transitiw, S Translate modulo, l, 83 Translation invariance, S4 Unbounded abow,10 below,10 func:tfon, interval, 7 sot, S
Uncountable set, 28 Uncountably infinite set, 28
Uniform continuous; 13 CODvel'JODCO almost, 112 Union, 2 Upper bound, s left dorivatiw,170 limit, tO risht derivatiw, 170
Variation .tun~.182
·bounded,179 nesatiw. 183 positiw,-183 total,180
Vitali cover.173 c:overina thoorem.173 Void set, 2 Weakly ccmveraent. 226 Well defined, 1
